A method for constrained energy minimization of macromolecules

Two algorithms for the local energy minimization of the structure of macromolecules in the presence of constraints are proposed. They are a combination of the method of steepest descents and the method of conjugate gradients with the procedure SHAKE, by which distance constraints can be satisfied. The two algorithms are tested by applying them to a small protein, the bovine pancreatic trypsin inhibitor (BPTI), and compared with the penalty function method for conserving constraints. The efficiency of the proposed methods depends on the level of interdependence of the constraints. For bond-length constraints, the use of SHAKE is superior to the penalty function method. However, when bond-angle constraints are included, SHAKE is more efficient only if the curvature of the penalty function is considerably greater than that of the potential function being minimized. The results indicate that with bond-length constraints the minimization behavior is similar to that without constraints. However, the simultaneous application of bond-length and bond-angle constraints appears to confine the molecule to a very limited part of configuration space, very different from the part covered by an unconstrained minimization. This conclusion calls into question energy minimizations of protein systems in which only the dihedral angles are allowed to vary.     

Analysis of an energy minimization method for locating transition states on potential energy hypersurfaces

Conditions are given for the successful search for a transition state by an energy minimization method. Proofs for these guidelines are presented. Advantages of this method are discussed, including its use in establishing lower bounds to transition state energies. Comparisons are made with other searching methods.     

Newton leaves on potential energy surfaces

The reaction path (RP) is an important concept of theoretical chemistry. We generalize the definition of the Newton trajectory (NT), as an RP, to Newton leaves in a higher dimensional subspace of the configuration space. Our standpoint is that of Bofill and Anglada [(2001) Theor. Chem. Acc. 105:436], who used a reduced potential energy surface for finding an RP. An NT follows an RP curve where the gradient is always a pointer to a fixed direction. More generally, a Newton leaf is a subspace of coordinates where the gradient can move in a subspace of directions. We report some known mathematical properties of Newton leaves. We explain the construction of Newton leaves with the example of a 3D test surface in ⁴ [W.Quapp et al. (1998) Theor. Chem. Acc. 100:285], because three coordinate dimensions are the smallest number of dimensions one needs at least to understand a Newton leaf in contrast to the known NTs.Acknowledgement The work was made possible through financial support of the Deutsche Forschungsgemeinschaft. The authors thank D. Heidrich for stimulating discussions.     

A truncated Newton minimizer adapted for CHARMM and biomolecular applications

We report the adaptation of the truncated Newton minimization package TNPACK for CHARMM and biomolecular energy minimization. TNPACK is based on the preconditioned linear conjugate–gradient technique for solving the Newton equations. The structure of the problem—sparsity of the Hessian—is exploited for preconditioning. Experience with the new version of TNPACK is presented on a series of molecular systems of biological and numerical interest: alanine dipeptide (N-methyl-alanyl-acetamide), a dimer of N-methyl-acetamide, deca-alanine, mellitin (26 residues), avian pancreatic polypeptide (36 residues), rubredoxin (52 residues), bovine pancreatic trypsin inhibitor (58 residues), a dimer of insulin (99 residues), and lysozyme (130 residues). Detailed comparisons among the minimization algorithms available in CHARMM, particularly those used for large-scale problems, are presented along with new mathematical developments in TNPACK. The new TNPACK version performs significantly better than ABNR, the most competitive minimizer in CHARMM, for all systems tested in terms of CPU time when curvature information (Hessian/vector product) is calculated by a finite-difference of gradients (the numeric option of TNPACK). The remaining derivative quantities are, however, evaluated analytically in TNPACK. The CPU gain is 50% or more (speedup factors of 1.5 to 2.5) for the largest molecular systems tested and even greater for smaller systems (CPU factors of 1 to 4 for small systems and 1 to 5 for medium systems). TNPACK uses curvature information to escape from undesired configurational regions and to ensure the identification of true local minima. It converges rapidly once a convex region is reached and achieves very low final gradient norms, such as of order 10^?8, with little additional work. Even greater overall CPU gains are expected for large-scale minimization problems by making the architectures of CHARMM and TNPACK more compatible with respect to the second-derivative calculations. © 1994 by John Wiley & Sons, Inc.     

A stable,rapidly converging conjugate gradient method for energy minimization

We apply Shanno's conjugate gradient algorithm to the problem of minimizing the potential energy function associated with molecular mechanical calculations. Shanno's algorithm is stable with respect to roundoff errors and inexact line searches and converges rapidly to a minimum. Equally important, this algorithm can improve the rate of convergence to a minimum by a factor of 5 relative to Fletcher-Reeves or Polak-Ribière minimizers when used within the molecular mechanics package AMBER. Comparable improvements are found for a limited number of simulations when the Polak-Ribière direction vector is incorporated into the Shanno algorithm.     

Newton trajectories for finding stationary points on molecular potential energy surfaces

We present a new algorithm for computing Newton trajectories based on the Quadratic String Method (QSM) and explain how this can be used to find key stationary points on the molecular potential energy surface (PES). This method starts by using the intersections of Newton trajectories to locate stationary points on the PES. These points could then be used to determine the minimum energy path. The new method, called QSM-NT, is shown to be efficient and reliable for both analytical potential energy surfaces and potential energy surfaces computed from quantum chemistry calculations. The advantages and pitfalls of this method for exploring PES are discussed. In particular, the problem of discontinuous Newton trajectories is elucidated.     

B-spline method for energy minimization in grid-based molecular mechanics calculations

A method is described for molecular mechanics calculations based on a cubic B-spline approximation of the potential energy. This method is useful when parts of the system are allowed to remain fixed in position so that a potential energy grid can be precalculated and used to approximate the interaction energy between parts of a molecule or between molecules. We adapted and modified the conventional B-spline method to provide an approximation of the Empirical Conformational Energy Program for Peptides (ECEPP) potential energy function. The advantage of the B-spline method over simpler approximations is that the resulting B-spline function is C2 continuous, which allows minimization of the potential energy by any local minimization algorithm. The standard B-spline method provides a good approximation of the electrostatic energy; but in order to reproduce the Lennard–Jones and hydrogen-bonding functional forms accurately, it was necessary to modify the standard B-spline method. This modification of the B-spline method can also be used to improve the accuracy of trilinear interpolation for simulations that do not require continuous derivatives. As an example, we apply the B-spline method to rigid-body docking energy calculations using the ECEPP potential energy function. Energies are calculated for the complex of Phe-Pro-Arg with thrombin. For this system, we compare the performance of the B-spline method to that of the standard pairwise summation in terms of speed, accuracy, and overhead costs for a variety of grid spacings. In our rigid-body docking calculations, the B-spline method provided an accurate approximation of the total energy of the system, and it resulted in an 180-fold reduction in the time required for a single energy and gradient calculation for this system. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 71–85, 1998     

Global optimization using bad derivatives: Derivative-free method for molecular energy minimization

A general method designed to isolate the global minimum of a multidimensional objective function with multiple minima is presented. The algorithm exploits an integral “coarse-graining” transformation of the objective function, U, into a smoothed function with few minima. When the coarse-graining is defined over a cubic neighborhood of length scale ϵ, the exact gradient of the smoothed function, 𝒰_ϵ, is a simple three-point finite difference of U. When ϵ is very large, the gradient of 𝒰_ϵ appears to be a “bad derivative” of U. Because the gradient of 𝒰_ϵ is a simple function of U, minimization on the smoothed surface requires no explicit calculation or differentiation of 𝒰_ϵ. The minimization method is “derivative-free” and may be applied to optimization problems involving functions that are not smooth or differentiable. Generalization to functions in high-dimensional space is straightforward. In the context of molecular conformational optimization, the method may be used to minimize the potential energy or, preferably, to maximize the Boltzmann probability function. The algorithm is applied to conformational optimization of a model potential, Lennard–Jones atomic clusters, and a tetrapeptide. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1445–1455, 1998     

An efficient newton-like method for molecular mechanics energy minimization of large molecules

Techniques from numerical analysis and crystallographic refinement have been combined to produce a variant of the Truncated Newton nonlinear optimization procedure. The new algorithm shows particular promise for potential energy minimization of large molecular systems. Usual implementations of Newton's method require storage space proportional to the number of atoms squared (i.e., O(N²)) and computer time of O(N³). Our suggested implementation of the Truncated Newton technique requires storage of less than O(N^1.5) and CPU time of less than O(N²) for structures containing several hundred to a few thousand atoms. The algorithm exhibits quadratic convergence near the minimum and is also very tolerant of poor initial structures. A comparison with existing optimization procedures is detailed for cyclohexane, arachidonic acid, and the small protein crambin. In particular, a structure for crambin (662 atoms) has been refined to an RMS gradient of 3.6 × 10^?6 kcal/mol/Å per atom on the MM2 potential energy surface. Several suggestions are made which may lead to further improvement of the new method.     

A physically meaningful method for the comparison of potential energy functions

In the study of the conformational behavior of complex systems, such as proteins, several related statistical measures are commonly used to compare two different potential energy functions. Among them, the Pearson's correlation coefficient r has no units and allows only semiquantitative statements to be made. Those that do have units of energy and whose value may be compared to a physically relevant scale, such as the root-mean-square deviation (RMSD), the mean error of the energies (ER), the standard deviation of the error (SDER) or the mean absolute error (AER), overestimate the distance between potentials. Moreover, their precise statistical meaning is far from clear. In this article, a new measure of the distance between potential energy functions is defined that overcomes the aforementioned difficulties. In addition, its precise physical meaning is discussed, the important issue of its additivity is investigated, and some possible applications are proposed. Finally, two of these applications are illustrated with practical examples: the study of the van der Waals energy, as implemented in CHARMM, in the Trp-Cage protein (PDB code 1L2Y) and the comparison of different levels of the theory in the ab initio study of the Ramachandran map of the model peptide HCO-L-Ala-NH2.     

Gradient SHAKE: An improved method for constrained energy minimization in macromolecular simulations

Current macromolecular energy minimization algorithms become inefficient and prone to failure when bond length constraints are imposed. They are required to relieve steric stresses in biomolecules prior to a molecular dynamics simulation. Unfortunately, the latter often require constraints, leading to difficulties in initiating trajectories from unconstrained energy minima. This difficulty was overcome by requiring that the components of the energy gradient vanish along the constrained bonds. The modified energy minimization algorithm converges to a lower energy in a fewer number of iterations and is more robust than current implementations. The method has been successfully applied to the Dickerson DNA dodecamer, CGCGAATTCGCG. © 1995 John Wiley & Sons, Inc.     

Modelling of potential energy curves for diatomics: Reference molecular potential method

A method is suggested for modelling the potential energy curves of diatomics. It involves a search for the mapping which transfers the structural details from an analytically prescribed reference potential. For the ground-state potentials of seven covalent molecules, the accuracy in describing the spectroscopic region is comparable with the best empirical potentials, and the correct asymptotic behaviour at R → 0 and R → ∞ are also obtained.     

Global energy minimization by rotational energy embedding

Given a sufficiently good empirical potential function for the internal energy of molecules, prediction of the preferred conformations is nearly impossible for large molecules because of the enormous number of local energy minima. Energy embedding has been a promising method for locating extremely good local minima, if not always the global minimum. The algorithm starts by locating a very good local minimum when the molecule is in a high-dimensional Euclidean space, and then it gradually projects down to three dimensions while allowing the molecule to relax its energy throughout the process. Now we present a variation on the method, called rotational energy embedding, where the descent into three dimensions is carried out by a sequence of internal rotations that are the multidimensional generalization of varying torsion angles in three dimensions. The new method avoids certain kinds of difficulties experienced by ordinary energy embedding and enables us to locate conformations very near the native for avian pancreatic polypeptide and apamin, given only their amino acid sequences and a suitable potential function.     

A powerful method to prepare sulfur-rich macrocycles

Various cyclic polysulfanes (up to a 16-membered ring) have been selectively obtained from the corresponding disubstituted trityl-protected polysulfane benzene derivatives upon treatment with elemental iodine in the presence of silica gel. Depending on the position of the sulfur-rich moieties and the presence of a methylene linker between the benzene ring and sulfur atoms, two types of products containing either one or two benzene subunits have been isolated. The scope of the reaction, its selectivity, and mechanism are discussed.     

Empirical energy functions for energy minimization and dynamics of nucleic acids

An improved empirical energy function for energy minimization and dynamics calculations of nucleic acids is developed and evaluated by an examination of its representation of both static and dynamic properties of model systems. Among the properties studied and used for parameter optimization are base pairing interactions, sugar and phosphate energy surfaces, small crystal heats of sublimation, base, phosphate and sugar analogue vibration spectra, and the overall behavior of a DNA hexamer duplex in vacuum molecular dynamics simulations. The results obtained are compared with those from two other energy functions that have been used recently for nucleic acids. Parameters for two energy functions are given; one includes heavy atoms and only polar hydrogens and the other includes all atoms.     

A comparison of accelerators for direct energy minimization in electronic structure calculations

We compare three different methods for direct energy minimization in electronic structure calculations where the gradient of the energy functional with respect to the molecular orbitals is available. These methods make use of the preconditioned gradient to increase robustness. An orbital transformation is used to ensure that the orthogonality constraint on the orbitals remains satisfied when using standard minimization methods. In addition, we propose an adaptive scheme for estimating the curvature of the energy functional to increase the performance of a line search free quasi-Newton method. We show that the performance of all methods is similar when robustness of the methods is ensured.     

A rapid and efficient algorithm for packing polypeptide chains by energy minimization

An improved algorithm for packing polypeptide chains with fixed geometry, which converges to a local energy minimum rapidly and efficiently, is described. The speed of convergence of the new algorithm is comparable to that of existing algorithms for minimizing the energies of single polypeptide chains, and it is several times greater than the speed of convergence of previous algorithms for minimizing the energy of structures consisting of several polypeptide chains. The algorithm has been used to minimize the energy of three-stranded (L -Ala)₈ β-sheets, three-stranded (L -Val)₆ β-sheets, and five-stranded (L -Ile)₆ β-sheets, starting from regular structures found previously; of the three-stranded regular and truncated (Gly-L -Pro-L -Pro)₄ structures used in earlier work to model collagen; and of the stacked β-sheet (L -Ala-GLy)₆ structures used to model silk. The antiparallel L -Ala β-sheet, and Gly-Pro-Pro triple helices, and the silk II structure remained nearly regular after energy minimization, but by contrast with results from earlier computations the other structures became significantly irregular. © 1994 by John Wiley & Sons, Inc.     

The conjugate gradient method applied to the direct second-order minimization of energy

The conjugate gradient method is proposed for minimizing the second-order Δε. Formulae for derivatives are expressed in a constructive way: they specify to which derivatives and with which weight each integral (belonging to a non-redundant list) contributes. The case of geminal functions is fully treated. A test calculation on the BeH molecule is shown.     

A reduced-restricted-quasi-Newton–Raphson method for locating and optimizing energy crossing points between two potential energy surfaces

We present a method for the location and optimization of an intersection energy point between two potential energy surfaces. The procedure directly optimizes the excited state energy using a quasi-Newton–Raphson method coupled with a restricted step algorithm. A linear transformation is also used for the solution of the quasi-Newton–Raphson equations. The efficiency of the algorithm is analyzed and demonstrated in some examples. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 :992–1003, 1997